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line segment
Definition Suppose $V$ is a vector space over $\mathbb{R}$ or $\mathbb{C}$, and $L$ is a subset of $V$. Then $L$ is a line segment if $L$ can be parametrized as
$L=\{a+tb\mid t\in[0,1]\}$ 
for some $a,b$ in $V$ with $b\neq 0$.
Sometimes one needs to distinguish between open and closed line segments. Then one defines a closed line segment as above, and an open line segment as a subset $L$ that can be parametrized as
$L=\{a+tb\mid t\in(0,1)\}$ 
for some $a,b$ in $V$ with $b\neq 0$.
If $x$ and $y$ are two vectors in $V$ and $x\neq y$, then we denote by $[x,y]$ the set connecting $x$ and $y$. This is , $\{\alpha x+(1\alpha)y\ 0\leq\alpha\leq 1\}$. One can easily check that $[x,y]$ is a closed line segment.
Remarks

An alternative, equivalent, definition is as follows: A (closed) line segment is a convex hull of two distinct points.

If $V$ is a topological vector space, then a closed line segment is a closed set in $V$. However, an open line segment is an open set in $V$ if and only if $V$ is onedimensional.

More generally than above, the concept of a line segment can be defined in an ordered geometry.
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